Skip to content
All library documents

Quantiles of Fractional Brownian Motion from Gaussian Marginals

Article Quant Q&A · Author: Serg Gini
Read the original

Summary

The document asks whether a quantile function can be calculated for fractional Brownian motion (fBm), drawing a comparison with geometric Brownian motion, whose quantiles can be expressed using the standard normal quantile function. The question notes that fBm is described in cited material as a centered Gaussian process, but asks whether that property makes a quantile function applicable.

The central statistical issue is the distinction between quantiles of a process at a fixed time and quantiles of an entire path. Gaussianity gives a normal marginal distribution at each fixed time, which supports computing pointwise quantiles from that marginal’s mean and variance. It does not by itself provide a single quantile function for the whole dependent path. The document contains no answer or derivation, so it leaves the exact construction and treatment of path-level quantiles unresolved. Any application would need to specify the time point or path statistic of interest.

Key ideas

  • Fractional Brownian motion is described in the question as a centered Gaussian process.
  • A process’s marginal distribution at a fixed time can have quantiles even when its observations are dependent over time.
  • The quantile of a whole path is distinct from a pointwise quantile.
  • The document poses the question but does not provide a derivation or answer.

Tags

StatisticsVolatility
Full text
# Quantile function for fractional Brownian motion (fBm)


# Quantile function for fractional Brownian motion (fBm)












If anyone could help me to understand if it is possible calculate the quantile function for fBm?

I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised Gaussian, this is not clear for me if the quantile function is applicable here.

Based on point 7 here: https://www.randomservices.org/random/brown/Geometric.html It is possible to write quantile GBM function with the standard normal quantile function.

Is it possible for fBm?

Thank you in advance.

[1] http://www.columbia.edu/~ad3217/fbm/thesisold.pdf

[2] https://www.diva-portal.org/smash/get/diva2:1257290/FULLTEXT01.pdf

[3] https://www.sciencedirect.com/science/article/pii/S111001682030541X

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.